Zero element

In mathematics, a zero element is one of several generalizations of the number zero to other algebraic structures. These alternate meanings may or may not reduce to the same thing, depending on the context.

Contents 1 Additive identities 2 Absorbing elements 3 Zero objects 4 Zero morphisms 5 Least elements

Additive identities

An additive identity is the identity element in an additive group. It generalises the property 0 + x = x. Examples include:

• The null vector under vector addition
• The zero function or zero map, defined by z(x) = 0, under function addition, (f + g)(x) = f(x) + g(x), since z + f = f.
• The empty set under set union
• An empty sum or empty coproduct
• An initial object in a category (an empty coproduct, and so an identity under coproducts)

Absorbing elements

An absorbing element in a multiplicative semigroup or semiring generalises the property 0 × x = 0. Examples include:

• The empty set, which is an absorbing element under Cartesian product of sets, since {} × S = {}
• The zero function or zero map, defined by z(x) = 0, under function multiplication, (f × g)(x) = f(x) × g(x), since z × f = z.

Many absorbing elements are also additive identities, including the empty set and the zero function. Another important example is the distinguished element 0 in a field or ring, which is both the additive identity and the multiplicative absorbing element, and whose principal ideal is the smallest ideal.

Zero objects

A zero object in a category is both an initial and terminal object (and so an identity under both coproducts and products). For example, the trivial structure (containing only the identity) is a zero object in categories where morphisms must map identities to identities. Specific examples include:

• The trivial group, containing only the identity (a zero object in the category of groups)
• The zero module, containing only the identity (a zero object in the category of modules over a ring)

Zero morphisms

A zero morphism in a category is a generalised absorbing element under function composition: any morphism composed with a zero morphism gives a zero morphism. Specifically, if 0_XY : X → Y is the zero morphism among morphisms from X to Y, and f : A → X and g : Y → B are arbitrary morphisms, then g ∘ 0_XY = 0_XB and 0_XY ∘ f = 0_AY.

If a category has a zero object 0, then there are canonical morphisms X → 0 and 0 → Y, and composing them gives a zero morphism 0_XY : X → Y. In the category of groups, for example, zero morphisms are morphisms which always return group identities, thus generalising the function z(x) = 0.

Least elements

A least element in a partially ordered set or lattice may sometimes be called a zero element, and written either as 0 or ⊥.

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